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17. Let G be a Lie group and let g be its Lie algebra. The Weil algebra of G is the bigraded algebra W (G) defined by W p,q (G) = W p,q (G, R) = S p (g) ⊗ E q−p (g). Initially we will consider the Weil algebra only as a graded algebra with the total degree. The map f∇ : E ∗ (g) → E ∗ (E, R) can be extended to a map defined on W (G). To this end, for x ∈ S 1 (g) = g∨ , we write f∇ (x) = φ(x) and we extend f∇ multiplicatively. By definition f∇ is a morphism of algebras. 3). We first define the operator i(h).
With the hypotheses of the above proposition, there is a natural isomorphism ∗ Hcont (G, R) → H ∗ (g, u, R). 3. Computation of Continuous Cohomology The van Est isomorphism can be used to compute the continuous cohomology of Lie groups. Let G be an algebraic reductive group defined over R. Let us assume that G(R) is a connected Lie group. Then G(C) is a complex connected reductive group. Let gR be the Lie algebra of G(R). Then gC = g ⊗ C is the Lie algebra of G(C). Let K be a maximal compact subgroup of G(R), let k be the Lie algebra of K and let gR = k ⊕ p be the Cartan decomposition of gR with respect to k.
Let Erp,q be the Leray spectral sequence of the bundle S. Since it is a S 2n−1 -bundle, then Erp,q = 0 for q = 0, 2n − 1. Thus the only nonzero differential is d2n . As in the previous section, the standard orientation of F as complex vector bundle defines a class σ2n+1 ∈ H 2n−1 (S 2n−1 , Z) = E20,2n−1 . The Euler class of F is the class e(F ) = d2n σ2n−1 ∈ E 2n,0 = H 2n (X, Z). To give an inductive definition of Chern classes we need two more facts. First, observe that, for j < 2n − 1, the morphism πS∗ : H j (X, Z) → H j (S, Z) is an isomorphism.